What is the value of (cosec `60^(@)-(1)/(2))` ?

Options

A. `((4-sqrt(3)))/(2sqrt(3))`
B. `((2sqrt(3)-1))/(sqrt(3))`
C. `((3sqrt(3)-1))/(3)`
D. `(1)/(sqrt(3))`

Answer

To solve the question \( \csc(60^\circ) - \frac{1}{2} \), we will follow these steps:

### Step 1: Find the value of \( \csc(60^\circ) \)
The cosecant function is the reciprocal of the sine function. Therefore, we need to find \( \sin(60^\circ) \) first.

\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

Now, we can find \( \csc(60^\circ) \):

\[ \csc(60^\circ) = \frac{1}{\sin(60^\circ)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 2: Substitute \( \csc(60^\circ) \) into the expression
Now, we substitute \( \csc(60^\circ) \) into the expression:

\[ \csc(60^\circ) - \frac{1}{2} = \frac{2}{\sqrt{3}} - \frac{1}{2} \]

### Step 3: Find a common denominator
To subtract these two fractions, we need a common denominator. The least common multiple of \( \sqrt{3} \) and \( 2 \) is \( 2\sqrt{3} \).

### Step 4: Rewrite each term with the common denominator
Now we rewrite each term:

\[ \frac{2}{\sqrt{3}} = \frac{2 \cdot 2}{\sqrt{3} \cdot 2} = \frac{4}{2\sqrt{3}} \]

\[ \frac{1}{2} = \frac{1 \cdot \sqrt{3}}{2 \cdot \sqrt{3}} = \frac{\sqrt{3}}{2\sqrt{3}} \]

### Step 5: Subtract the fractions
Now we can subtract the two fractions:

\[ \frac{4}{2\sqrt{3}} - \frac{\sqrt{3}}{2\sqrt{3}} = \frac{4 - \sqrt{3}}{2\sqrt{3}} \]

### Final Answer
Thus, the value of \( \csc(60^\circ) - \frac{1}{2} \) is:

\[ \frac{4 - \sqrt{3}}{2\sqrt{3}} \]

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